Now, following [1]}, we have to
decouple the time-like direction and to present the 3D extrinsic curvature \(K_{ab}\)  in terms of the 2D extrinsic curvature \(k_{AB}\)  of the normal to \(u^a\)  hypersurface. Clearly, the metric on this hypersurface is a \(4 \times 4\)  matrix \(\gamma _{ab}\) , the only non-trivial part of which is the \(2\times 2\)  block \(\gamma _{AB}\)  determined by (REF ). Details on the relation between 3D and 2D extrinsic curvatures can be find in Appendix A. Here we just recall that \(K_{ab}=\alpha ^{-1}\left(k_{ab}-g_{{\cal H}} u_a u_b \right)+\Omega _a u_b+\Omega _b u_a\) , where \(\alpha \)  is the renormalization factor making divergent quantities finite on the event horizon, \(\Omega _a\)  is the Hájiček field and \(g_{{\cal H}}\)  is the surface gravity. (Actually, we will not use this relation in what follows, so the explicit form of the Hájiček field does not matter.)
